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arXiv:2001.02645 · Physical review. B./Physical review. B

Excitations with projected entangled pair states using the corner transfer matrix method

Boris Ponsioen, Philippe Corboz

Published May 5, 2020• 54 citations
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Abstract

We present an extension of a framework for simulating single quasiparticle or collective excitations on top of strongly correlated quantum many-body ground states using infinite projected entangled pair states, a tensor network ansatz for two-dimensional wave functions in the thermodynamic limit. Our approach performs a systematic summation of locally perturbed states in order to obtain excited eigenstates localized in momentum space, using the corner transfer matrix method, and generalizes the framework to arbitrary unit cell sizes, the implementation of global Abelian symmetries and fermionic systems. Results for several test cases are presented, including the transverse Ising model, the spin-$1/2$ Heisenberg model, and a free fermionic model, to demonstrate the capability of the method to accurately capture dispersions. We also provide insight into the nature of excitations at the $k=(\ensuremath{\pi},0)$ point of the Heisenberg model.

DOI
10.1103/physrevb.101.195109

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We present an extension of a framework for simulating single quasiparticle or collective excitations on top of strongly correlated quantum many-body ground states using infinite projected entangled pair states, a tensor network ansatz for two-dimensional wave functions in the thermodynamic limit. Our approach performs a systematic summation of locally perturbed states in order to obtain excited eigenstates localized in momentum space, using the corner transfer matrix method, and generalizes the framework to arbitrary unit cell sizes, the implementation of global Abelian symmetries and fermionic systems. Results for several test cases are presented, including the transverse Ising model, the spin-1/21/21/2 Heisenberg model, and a free fermionic model, to demonstrate the capability of the method to accurately capture dispersions. We also provide insight into the nature of excitations at the k=(\ensuremathπ,0)k=(\ensuremath{\pi},0)k=(\ensuremathπ,0) point of the Heisenberg model.
Numerical treatment of two dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this work, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated J1−J2J_1-J_2J1​−J2​ Heisenberg model illustrate the improved precision, efficiency and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.
We develop and benchmark a technique for simulating excitation spectra of generic two-dimensional quantum lattice systems using the framework of projected entangled-pair states (PEPS). The technique relies on a variational ansatz for capturing quasiparticle excitations on top of a PEPS ground state. Our method perfectly captures the quasiparticle dispersion relation of the square-lattice transverse-field Ising model and reproduces the spin-wave velocity and the spin-wave anomaly in the square-lattice Heisenberg model with high precision.
Numerical treatment of two-dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground-state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this paper, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model illustrate the improved precision, efficiency, and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.
Accurate computation of the energy variance and ⟨⟨L†L⟩⟩\langle\langle \mathcal{L}^\dagger \mathcal{L} \rangle\rangle⟨⟨L†L⟩⟩ using iPEPS
Infinite projected entangled-pair states (iPEPS) provide a powerful tensor network ansatz for two-dimensional quantum many-body systems in the thermodynamic limit. In this paper we introduce an approach to accurately compute the energy variance of an iPEPS, enabling systematic extrapolations of the ground-state energy to the exact zero-variance limit. It is based on the contraction of a large cell of tensors using the corner transfer matrix renormalization group (CTRMG) method, to evaluate the correlator between pairs of local Hamiltonian terms. We show that the accuracy of this approach is substantially higher than that of previous methods, and we demonstrate the usefulness of variance extrapolation for the Heisenberg model, for a free fermionic model, and for the Shastry-Sutherland model. Finally, we apply the approach to compute ⟨⟨L†L⟩⟩\langle \langle \mathcal{L}^\dagger \mathcal{L} \rangle \rangle⟨⟨L†L⟩⟩ for an open quantum system described by the Liouvillian L\mathcal{L}L, in order to assess the quality of the steady-state solution and to locate first-order phase transitions, using the dissipative quantum Ising model as an example.
Infinite projected entangled-pair states (iPEPS) provide a powerful tool to study two-dimensional strongly correlated systems directly in the thermodynamic limit. In this work, we extend the iPEPS toolbox by a method to efficiently evaluate nonequal time two-point correlators, enabling the computation of spectral functions. It is based on an iPEPS ansatz of the ground state in a large unit cell, with an operator applied in the center of the cell, which is evolved in real-time using the fast-full update method. At every time step, the two-point correlators within a cell are computed based on the corner transfer matrix renormalization group method. Benchmark results for the two-dimensional transverse field Ising model show that the main features of the dynamical structure factor can already be reproduced at relatively small bond dimensions and unit cell sizes. The results for the magnon dispersion are found to be in good agreement with previous data obtained with the iPEPS excitation ansatz.
The excitation ansatz for tensor networks is a powerful tool for simulating the low-lying quasiparticle excitations above ground states of strongly correlated quantum many-body systems. Recently, the two-dimensional tensor network class of infinite projected entangled-pair states gained new ground state optimization methods based on automatic differentiation, which are at the same time highly accurate and simple to implement. Naturally, the question arises whether these new ideas can also be used to optimize the excitation ansatz, which has recently been implemented in two dimensions as well. In this paper, we describe a straightforward way to reimplement the framework for excitations using automatic differentiation, and demonstrate its performance for the Hubbard model at half filling.